8 LEBESGUE'S CONVERGENCE THEOREMS 129
from which it follows ((3.16.5) and (8.7.8)) that, for almost all x, the function
f(x, z) is the limit of the sequence I/T1 Y,fkn(x)}9 where
\ fc-i /
_/(*, a + re2kni/n)re2kni/n
Jkn\XJ "" !------2kni/n-----------"" *
a + re2kni/n — z
Applying (13.8.4), it follows that h(z) is the limit of the sequence
and therefore that
which shows that h is analytic in E (9.9.4). By replacing the term !/(( — z) in
the right-hand side of (188.8.131.52) by fc!/(f-z)*+1, we obtain the formula
stated for D*/z(z).
(13.8.7) (i) If (An) is a sequence of integrable sets, then A = (°) An w
integrable. If also the sequence (Aw) w decreasing, we have
(184.108.40.206) MA) = mfXAJ.
(ii) If(An) is a sequence of integrable sets contained in an integrable set B,
then A = (J AM w integrable.
(iii) If(An) is a sequence of integrable sets such that
/on A = (J Art is integrable, and we have
If moreover the sets An are pairwise disjoint, then
(" complete additivity of measure *'). the right-hand side of (220.127.116.11). But it follows from the hypotheses